276 HlSTOBY OP THE THEOEY OF NUMBEES. [CHAP. IX He stated empirically that N is the number of reduced forms ay2+byz+csf, 4ac-~&2=p for b odd, ac— £&2=p for b even, where b<a, b<c. C. F. Arndt115 proved in two ways that the product of all integers relatively prune to M~pn or 2pn, and not exceeding (M— 1)/2, is ==«=! (mod M), when p is a prime 4fc+3, the sign being + or — according as the number of residues >M/2 of M is even or odd. Again, (l-3-5-7...(p-2)}2==fcl (modp), the sign being + or — according as the prune p is of the form 4n+3 or 4n+l. In the first case, 1- 3 ... (p —2)= ± 1 (mod p). L. Kronecker116 obtained, for Dirichlet's113 exponent ?#, the result m^v (mod 2), where y is the number of positive integers of the form q*l+lr2 in the set p— 22, p— 42, p — 62, . . ., and q is a prime not dividing r. Liou-ville (p. 267) gave m=&-f v* (mod 2), when p = 8&+3 and v" is the number of positive integers of the form #4H"V2 in the set p— 42, p— 82, p — 122, — J. Liouville117 gave the result m=<r+r (mod 2), for the case p = 8&+3, where r is the number of positive integers of the form 2g4*4"1 r2 (q a prune not dividing r) in the set p — I2, p — 32, p— 52, . . ., and cr is the number of equal or distinct prunes 40+1 dividing 6, where p = a2+262 (uniquely). A. Korkine118 stated that, it [x] is the greatest integer ^x, — Q (P-3)/4p - -i (modp). J. Franel119 proved the last result by use of Legendre's symbol and . (mod2)- M. Lerch120 obtauied Jacobi's114 result. H. S. Vandiver1200 proved Dirichlet's113 result and that (p-l)/2[-7-2-| = 2 K-y-i LPJ (mod 2). R. D. Carmichael121 noted that (4) holds if and only if p is a prime. E. Malo122 considered the residue ^r of 1-2.. .(p —1)/2 modulo p, where p is a prime 4ra+l, and 0<r<p/2. Thus r2s= —1. The numbers 2, 3,..., (p —1)/2, with r excluded, may be paired so that the product of the two of a pair is s==*=l (mod p). If this sign is minus for k pairs, 1-2.. .(p-l)/25=(-l)V (mod p). *J. Ouspensky gave a rule to find the sign in (5). OTHER CONGRUENCES INVOLVING FACTORIALS. V. Bouniakowsky129 noted that (p-l)!=PP', P±P'=0 (mod p) according as p=4&=Fl. For, if p is a primitive root of p, we may set P = pp2 "BArchiv Math. Phys., 2, 1842, 32, 34-35. 120°Amer. Math. Monthly, 11, 1904, 51-6. ^Jour. de Math., (2), 5, I860, 127. ™Ibid., 12, 1905, 106-8. ™Ibid., 128. J22L;mterm6diaire des math., 13, 1906, 131-2 118L'interm6diaire dea math., 1, 1894, 95. 123Bull. Soc. Phys. Math. Kasan, (2), 21. ™Ibid., 2, 1895, 35-37. 12«M6m. Ac. Sc. St. Ptesbourg, (6), 1, 1831, 564. i20Prag Sitzungsber. (Math.), 1898, No. 2.